LMIs in Control/Discrete-Time Systems/Discrete-Time Lyapunov Stability

Discrete-Time Lyapunov Stability

The System

Discrete-Time System

$\begin{array}{ccc}x(t{)}_{k+1}& =A_{d}x(t{)}_k,& A_d\in {𝐑}^{𝐧\mathbf{*}𝐧}\\ \end{array}$

The Data

The matrices $System{A}_d,P$.

The Optimization Problem

The following feasibility problem should be solved:

Find P obeying the LMI constraints.

The LMI:

Discrete-Time Lyapunov Stability

The LMI formulation

$\begin{array}{c}P\in {𝐒}^{𝐧}\\ Find& P>0,\\ \left[\begin{array}{c}{A}_d^{T}P{A}_d-P\end{array}\right]& \le 0\end{array}$

Conclusion:

If there exists a $P\in {𝐒}^{𝐧}$ satisfying the LMI then, $|{\lambda }_i(A_d)|\le 1,\mathrm{\forall }i=1,2,...,n;$ and the equilibrium point $\overline{x}=0$ of the system is Lyapunov stable.

Implementation

• [] - MATLAB implementation of the LMI.

Related LMIs

Continuous_Time_Lyapunov_Inequality - Lyapunov_Inequality

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.